Fourier integral of rectangular pulse

Fourier integral formula. For f(x) with ∫|f| dx finite:

f(x) = (1/π) ∫₀^∞ [ A(ω) cos ωx + B(ω) sin ωx ] dω

A(ω) = ∫{−∞}^{∞} f(v) cos ωv dv, B(ω) = ∫{−∞}^{∞} f(v) sin ωv dv.

Here f(x) = 1 for |x| < 1 and 0 for |x| > 1. It is an even function, so B(ω) = 0.

A(ω) = ∫₋₁¹ cos ωv dv = 2 ∫₀¹ cos ωv dv = 2 sin ω / ω

Fourier integral:

f(x) = (2/π) ∫₀^∞ (sin ω cos ωx / ω) dω

Convergence. The integral converges to f(x) wherever f is continuous, and to the average of the one-sided limits at a jump:

• |x| < 1: the integral = 1
• |x| > 1: the integral = 0
• x = ±1: the integral = ½

Consequence. At x = 0: (2/π) ∫₀^∞ sin ω/ω dω = 1, which gives the well-known result ∫₀^∞ (sin ω/ω) dω = π/2.